Eigenmodes of Photoacoustic T-Cells

Eigenmodes of Photoacoustic T-Cells Bernd Baumann1, Bernd Kost1, Hinrich Groninga2, Marcus Wolff1,2 1 University of Applied Sciences Hamburg, Mechanical and Production Engineering, [email protected], 2PAS-Tech GmbH, Zarrentin, [email protected] 1 Introduction The photoacoustic effect is based on resonant absorption of light by a sample and the transfer of the excitation energy into thermal energy via inelastic collisions of gas molecules. A modulated irradiation of the sample causes periodic pressure variations that can be detected by a microphone and measured using lock-in technique (Demtröder 2002). Photoacoustic spectroscopy finds many applications in the field of concentration measurements of gaseous compounds. The signal detection sensitivity of photoacoustic sensors strongly depends on the geometry of the photoacoustic cell. It can be considerably improved by taking advantage of acoustical cell resonances, i.e., the radiation is modulated at a frequency equivalent to an acoustical eigenmode of the measuring chamber. In order to optimize a photoacoustic system, it is key to precisely understand the distribution of pressure in the sample cell. Only then, it is possible to optimize the coupling of optical excitation, sound wave generation and microphone detection (Michaelian 2003). Therefore we have analysed the eigenmode structure of photoacoustic cells of different geometries using the finite element tool FEMLAB. Additionally, we have compared the resulting eigenfrequencies to experimentally determined eigenfrequencies. 2 Photoacoustic Sensors absorption cylinder: diameter DA = 26 mm length resonance cylinder: diminution: LA = 82 mm DR = 11 mm diameter length adjustable diameter DD = 8.9 mm length LD = 2 mm Fig. 1. Experimental set-up and dimensions of the photoacoustic sensor with T-cell. All dimensions are measured inside the cell. The most frequently used type of photoacoustic sensors is based on a cylinder shaped container (Michaelian 2003). In recent years the interest in photoacoustic cells with unconventional shape is increasing. A cell-type consisting of two intersecting cylinders has been proposed in (PAS-Tech 2004). The two cylinders form a T-geometry, consisting of an optical absorption cylinder and, centrically perpendicular to that, an acoustical resonance cylinder. This design allows independent optimization of the key parameters affecting the signal strength. At the end of the resonance cylinder the microphone is mounted. Figure 1 shows the experimental set-up and the dimensions of the photoacoustic sensor investigated in this paper. Measurements were performed on n -butane ( C4 H10 ) at atmospheric conditions ( 296 K , 1013 hPa ) resulting in a sound velocity of c = 212 m/s . Details concerning the experimental set-up can be found in (Wolff et al. 2005). 3 Acoustic Eigenmodes of T-cells Particularly in the context of acoustically resonant trace gas detection, the eigenmode structure of a photoacoustic cell is of importance. The problem can be solved analytically for cells of cylindrical shape. Therefore, we used cylindrical cells to verify the applicability of our numerical approach. The results of analytical and numerical calculations were found to be in excellent agreement. In this section, we investigate the modes of photoacoustic T-cells that don’t allow an analytical solution. Additionally, the finite element results for the eigenfrequencies of the T-cell are compared to experimental results. The eigenfrequencies, f i = equation ωi 2π , and the eigenmodes are obtained by solving the differential r2 r r 2 ∇ p(r ) + k p (r ) = 0, (1) r where p ( r ) is the deviation of the gas-pressure from its mean-value, k = ωc and c is the sound velocity. It is assumed, that the walls of the photoacoustic cell are sound-hard, which leads to the boundary condition ∂p =0 ∂n (2) (the normal derivative of the pressure is zero at the boundary). Tab. 1. Eigenfrequencies of the T-cell for different lengths LR of resonance cylinder. The frequencies for resonance and absorption cylinder are calculated using the formula for gas columns. The experimental frequencies are measured with an accuracy of about 1 %. FEM LR [mm] 320 160 80 40 20 10 f Resonance Cyl. Dev. Type f [Hz] [Hz] 197.9 497.9 813.7 354.0 953.2 1295.2 636.5 1295.2 1788.4 1114.6 1295.2 2583.2 1295.2 1824.2 2684.3 1295.2 2424.0 3124.0 165.6 496.9 828.1 331.3 993.8 [%] -16.3 -0.2 1.8 -6.4 4.3 662.5 4.1 1987.5 1325.0 11.1 18.9 3λ / 4 λ/4 2650.0 2.6 λ/2 Absorption Cyl. Dev. Type f [Hz] [%] -0.2 1292.7 2585.4 1292.7 1292.7 -0.2 -0.2 0.1 -0.2 -0.2 356 0.6 629 -1.2 1092 -2.0 1722 -5.6 2392 3003 -1.3 -3.9 λ/2 λ/4 1292.7 203 [%] 2.6 [Hz] λ/4 3λ / 4 5λ / 4 λ/4 3λ / 4 1292.7 Experiment Dev. f λ/2 λ/2 λ λ/2 λ/2 In the following, we present results for T-cells with different lengths LR of the resonance cylinder (see Table 1 and Figures 2 and 3). Resonance Cylinder 80 mm 160 mm 320 mm Absorption Cylinder Fig. 2. Pressure distribution of the three lowest nontrivial eigenmodes (1, 2, 3) of the T-cell along the symmetry axis of the absorption cylinder and the resonance cylinder for LR = 320 mm , LR = 160 mm and LR = 80 mm . The actual length of the abscissa in the figures on the right-hand side is DA + LD + LR (see Figure 1). The absolute values of the pressure is of no significance. Resonance Cylinder 10 mm 20 mm 40 mm Absorption Cylinder Fig. 3. Pressure distribution of the three lowest eigenmodes of the T-cell for LR = 40 mm , LR = 20 mm and LR = 10 mm . For details consult the caption of Figure 2. - LR = 320 mm : All three lowest eigenmodes (1-3) are mainly eigenmodes of the resonance cylinder (one end open, one end closed), see Figure 2. This interpretation is confirmed by Table 1 and the observation that the pressure values in the absorption cylinder are approximately zero (Figure 2, left). Only for the lowest eigenmode a minor deviation from zero is observed. The ground state (lowest eigenmode) has been observed experimentally and the deviation between measured frequency and FEMLAB result is small. - LR = 160 mm : The two lowest eigenmodes (1-2) are longitudinal modes of the resonance cylinder whereas the third mode is associated with the first longitudinal mode of the absorption cylinder (two closed ends). Because this mode shows a node at the opening of the resonance cylinder, it does not excite an oscillation in the resonance cylinder. Again, this interpretation is supported by Table 1 and the agreement between FEMLAB and measurement is excellent. - LR = 80 mm : The interpretation of the spectrum is essentially the same as in the case of the LR = 160 mm T-cell. Merely the order of the eigenmodes has changed: The first longitudinal mode of the absorption cylinder is now the second mode of the T-cell. The agreement with the measured frequency is good. - LR = 40 mm : The ground state represents the first longitudinal mode of the resonance cylinder. The second mode is the first longitudinal mode of the absorption cylinder (see Figure 3). The length of the resonance cylinder now almost equals the length of the absorption cylinder. Therefore, the third mode shows an interesting characteristic: The second longitudinal mode of the absorption cylinder is excited. This mode shows an antinode at the opening of the resonance cylinder. Therefore, a λ/ 2 -wave is excited in the resonance cylinder. That is, both the λ -wave of the absorption cylinder and the λ/ 2 -wave in the half-open resonance cylinder are observed. The latter would not be possible under usual conditions. The mode is depicted in Figure 4 (left). Agreement with the measured frequency is good. - LR = 20 mm : Now, the first longitudinal mode of the absorption cylinder constitutes the ground state of the T-cell. This mode cannot be detected experimentally, because of the node at the opening of the resonance cylinder. The experiment detected the remnants of the resonance cylinder’s first longitudinal mode. The length of the resonance cylinder has now become similar to the diameter of resonance and absorption cylinder. Therefore, the formula of the half-open cylinder is not applicable anymore. Figure 4 (right) shows, that the third mode cannot be disentangled into two cylinder modes. - LR = 10 mm : The ground state is again the first longitudinal mode of the absorption cylinder. The two higher modes cannot be brought into connection with cylinder modes. Both of these modes are detected experimentally. Fig.4. Third mode of LR = 40 mm -T-cell (left) and of LR = 20 mm -T-cell (right) . Depicted is p . For large lengths of the resonance cylinder one observes essentially modes, which are compatible with the well known formulae for the vibration of gas columns whereas for short resonance cylinders, this is no longer true. These observations are consistent with expectations and, therefore, we have a coherent picture of the T-cell’s spectrum. 4 Conclusions In this paper, we have demonstrated that the eigenmodes of photoacoustic cells of unconventional shape can be reliably determined using the finite element method. The agreement between finite element results and measured eigenfrequencies is good. However, not all of the numerically found eigenmodes of the T-cell have been detected experimentally. To understand which modes are observable, one has to consider the details of sound excitation and detection. The detection sensitivity of a photoacoustic sensor is strongly dependent to the energy loss of the sound wave. Therefore, loss has to be taken into account in order to model the photoacoustic signal strength. To do so, two possibilities exist: The first method is to include terms into the differential equation, which model the various loss mechanisms. The second approach includes the loss as a perturbation to the loss-free solution (Kreuzer 1977). The signal strength can then be obtained from the eigenmodes of the previous section. In addition to the determination of the eigenmodes it is necessary to calculate volume and surface integrals of certain quantities. FEMLAB offers the tools to calculate these integrals very fast. Since our future goal is to perform an automatic design optimization of photoacoustic cells, it is key to use efficient numerical methods. References Demtröder, W. (2002) Laser Spectroscopy, Springer Verlag, Berlin Kreuzer, L. B. (1977) The Physics of Signal Generation and Detection, Optoacoustic Spectroscopy and Detection, Pao, Y.-H. (Ed.), Academic, London: 1-25 Michaelian, K.H. (2003) Photoacoustic Infrared Spectroscopy, Wiley-Interscience, Hoboken PAS-Tech GmbH (2004) Verfahren zur Analyse und Konzentrationsmessung, DE 10 2004 053 480, Deutsches Patentamt, München Wolff, M.; Groninga, H.G.; Baumann, B.; Kost, B.; Harde, H. (2005) Resonance Investigations using PAS and FEM, Acta Acustica, Vol. 91, Suppl. 1, 99